Problem: Scott likes to run long distances. He can run $20 \text{ km}$ in $85$ minutes. He wants to know how many minutes $(m)$ it will take him to run $52 \text{ km}$ at the same pace. How long will it take Scott to run $52 \text{ km}$ ?
Explanation: We're dealing with a proportional relationship, so each ratio of kilometers to minutes must be equivalent. We can set up a proportion like this: $\dfrac{20 \text{ km}}{85 \text{ minutes}}=\dfrac{52 \text{ km}}{m \text{ minutes}}$ We can solve for $m$ by isolating it. Since the two rate expressions are equivalent, their reciprocals are also equivalent. $\begin{aligned} \dfrac{m \text{ minutes}}{52 \text{ km}}&=\dfrac{85 \text{ minutes}}{20 \text{ km}} \\\\ m \text{ minutes}&=\dfrac{85 \text{ minutes}}{20 \cancel{\text{ km}}}\cdot{52 \cancel{\text{ km}}} \end{aligned}$ $\begin{aligned} m&=\dfrac{85}{20}\cdot52 \\\\ &=\dfrac{17\cdot\cancel{5}}{\cancel{4}\cdot\cancel{5}}\cdot(13\cdot\cancel{4}) \\\\ &=17\cdot13 \\\\ &=221 \end{aligned}$ It will take Scott $221$ minutes to run $52 \text{ km}$.